Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces
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H erausgegeben von ]. L. Doob · A. Grothendieck · E. Heinz · F. Hirzebruch E. Hop£ · H. Hop£ · W. Maak · S. MacLane · W. Magnus M. M. Postnikov · F. K. Schmidt · D. S. Scott · K. Stein
Geschiiftsfiihrende H erausgeber B. Eckmann und B. L. van der Waerden
Ivan Singer
Best Approximation in N ormed Linear Spaces by Elements of Linear Subspaces
I Springer-Verlag Berlin Heidelberg GmbH 1970
Prof. Ivan Singer Academy of the Socialist Republic of Romania Institute of Mathematics, Bucharest
Geschiiftsfiihrende Herausgeber:
Prof. Dr. B. Eckmann Eidgeni:iss!sche Technische Hochschule Zurich
Prof. Dr. B. L. van der Waerden Mathematlsches Inst!tut der Un!vers!tiit Ziirioh
This monograph is a translation of the original Romanian version "Cea mai buna aproximare in spatii vectoriale normate prin elemente din subspatii vectoriale"
Translated by Radu Georgescu
ISBN 978-3-662-41585-6 ISBN 978-3-662-41583-2 (eBook) DOI 10.1007/978-3-662-41583-2
This work i> subject to copyright. AU rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publ!sher, the amount of the fee to be determined by agreement with the publisher
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by Springer-Verlag Berlin Hei:lelberg 1970.
Otiglnally published by Springer-Verlag Berlin Heidelberg New York in 1970. Softcover reptint of the hardcover 1st edition 1970
Library of Congress Catalog Card Number 73-110407 Title No. 5154
CONTENTS Page COXTENTS PREFACE .............................. .
5 9
PREFACE TO THE ENGLISH EDITION.
11
INTRODUCTION ............. .
13
Chapter I BEST APPROXIMATION IN NORMED UXEAR SPACES BY ELEMENTS OF ARBITRARY LINEAR SUBSPACES . . . . . . . . . . . . . ....................
17
§ 1. t:haracterizations or elements of best approximation . . . .
17
The first theorem of characterization of elements of best approximation in general normed linear spaces 1.2. Geometrical interpretation 1.3. Applications in the spaces C(Q) lA. Applications in the spaces Cn(Q) 1.5. Applications in the spaces L 1 ( T, v) 1.1). Applications in the spaces C\Q, v) and C~(Q, v) 1. 7. Applications in the spaces L 11 (1', 'J) (1 < p < oo) and in inner product spaces . . . . . . . . . . . . . . . . 1.8. The second theorem of characterization of elements of best approximation in general normed linear spaces 1.9. Geometrical interpretation . . . . . . . . . . . . . . 1.10. Applications and geometrical interpretation in the spaces 1.1.
C(Q) . . . . . . . . . . . . . . . . . . .
1.11. Applications in linear subspaces of the spaces C(Q) 1.12. Applications in the spaces L \ 1', v) . . . . . . . . 1.13. Other characterizations of elements of best approximation in general normed linear spaces . . . . . . . l.H. Orthogonality in general normed linear spaces §2. Existence of elements of best approxima
